Spurs have not lifted any silverware since they won the League Cup in 2008, when a Jonathan Woodgate header in extra-time saw them to victory over Chelsea.
And Redknapp believes that their drought could be set to continue, as the Lilywhites' current squad isn't as strong as others they have had in recent years.
"Tottenham need to start winning trophies," Redknapp said. "It's important for their history, when you look around you see pictures of photos from years gone by [of past successes] and that's what Spurs players want and what the fans demand.
"Are they getting closer, though? I'm not convinced. They have a lot of young players, and a strong nucleus, but they've had better teams over the past five, six years."
Tottenham's best chance of winning a trophy this season may have already gone, with the side falling out of the Capital One Cup to Arsenal.
Spurs lost 2-1 to their North London rivals on Wednesday night, and their only hope of silverware now comes in the Europa League and FA Cup.
Pochettino may argue that Redknapp is wrong to suggest that Spurs are not getting any closer to winning a trophy, though, as he did guide them to the League Cup final last season, where they lost 2-0 to Chelsea.
Tottenham are building for the future and although their current team may not be ready to take them to glory yet, the hope is that once their young players start to develop, they will be back competing for major honours before too long.